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Math Help - Finding Critical regions

  1. #1
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    Question Finding Critical regions

    Question:
    7. It is known from past records that 1 in 5 bowls produced in a pottery have minor defects. To monitor production a random sample of 25 bowls was taken and the number of such bowls with defects was recorded.

    (a) Using a 5% level of significance, find critical regions for a two-tailed test of the hypothesis that 1 in 5 bowls have defects. The probability of rejecting, in either tail, should be as close to 2.5% as possible.
    Answer:
    Critical regions is "X is equal and bigger than 1" or "X is equal to and bigger than 10"



    Basically, I do not understand how can "X is equal and bigger than 1" be one of the critical region?
    X~B(25, 0.2)

    I am thinking the P("X are equal and bigger than 1") = 0.0274 therefore, it goes over 2.5% hence, it shouldn't be the critical region??

    by the way, The probability for "X equal and bigger than 10" = 0.0173

    Could please you explain why one of the critical region is not P(X=0) which is 0.0038 instead of X less than and equal to 1?

    THank you.
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  2. #2
    Super Member
    Joined
    Sep 2008
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    Quote Originally Posted by Changhons View Post
    Question:
    7. It is known from past records that 1 in 5 bowls produced in a pottery have minor defects. To monitor production a random sample of 25 bowls was taken and the number of such bowls with defects was recorded.

    (a) Using a 5% level of significance, find critical regions for a two-tailed test of the hypothesis that 1 in 5 bowls have defects. The probability of rejecting, in either tail, should be as close to 2.5% as possible.
    Answer:
    Critical regions is "X is equal and bigger than 1" or "X is equal to and bigger than 10"



    Basically, I do not understand how can "X is equal and bigger than 1" be one of the critical region?
    X~B(25, 0.2)

    I am thinking the P("X are equal and bigger than 1") = 0.0274 therefore, it goes over 2.5% hence, it shouldn't be the critical region??

    by the way, The probability for "X equal and bigger than 10" = 0.0173

    Could please you explain why one of the critical region is not P(X=0) which is 0.0038 instead of X less than and equal to 1?

    THank you.
    Because the question, specifies as close to 2.5% as possible, P(X=0) is 0.38% were as for  P(X\leq 1 ) is 2.74%, and this value is closer to 2.5% ,if the question did not say as close to 2.5%, than you would look for a value in the tables that is less than or equal to 0.025, which in this case would be p(x=0)
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